However, restricting the inference to a low-dimensional latent space has the disadvantage that it admits no Lebesgue density for the posterior in the high-dimensional space of the actual variables [1]}. Furthermore, the estimation accuracy reached can strongly depend on the quality of the generative model [2]}. To circumvent this drawback and somehow escape from the lower dimensional manifold, [1]} takes the mean of the push-forward of the posterior in the latent space as a Bayes estimate. Some novel approaches also incorporate the inversion problem directly into the learning process [4]}, e.g. to achieve super resolution or to reconstruct high fidelity magnetic resonance images [5]}, [6]}. In this regard, we like to mention [7]} for their deep direct estimation procedure and the development of a conditional Wasserstein GAN discriminator, which allows sampling from the posterior.
